The Markus–Yamabe conjecture for continuous and discontinuous piecewise linear differential systems

Abstract

In 1960 Markus and Yamabe made the following conjecture: If a C 1 C^1 differential system x ˙ = F ( x ) \dot {\mathbf {x}}=F(\mathbf {x}) in R n \mathbb {R}^n has a unique equilibrium point and the Jacobian matrix of F ( x ) F(\mathbf {x}) for all x ∈ R n \mathbf {x}\in \mathbb {R}^n has all its eigenvalues with negative real part, then the equilibrium point is a global attractor. Until 1997 we do not have the complete answer to this conjecture. It is true in R 2 \mathbb {R}^2 , but it is false in R n \mathbb {R}^n for all n > 2 n>2 . Here we extend the conjecture of Markus and Yamabe to continuous and discontinuous piecewise linear differential systems in R n \mathbb {R}^n separated by a hyperplane, and we prove that for the continuous piecewise linear differential systems it is true in R 2 \mathbb {R}^2 , but it is false in R n \mathbb {R}^n for all n > 2 n>2 . But for discontinuous piecewise linear differential systems it is false in R n \mathbb {R}^n for all n ≥ 2 n\ge 2 .